To produce labels that won't deteriorate in the freezer go for a thermal printer. The grocery industry has used these for decades to produce labels that are affixed to the pallets that go into chilled and frozen storage. There will be no ink bleed, and no fading unless you plan to keep your venison frozen for 20 years or more!
I use one of these, a Brother QL-700:
QL-700 | High Speed Address Label Printer | Brother UK
I bought mine second-hand on eBay for 36 quid - there are normally plenty available.
It has proven to be excellent, with a variety of label sizes available along with bundled software (Brother's own P-Touch Editor) that allows easy design of attractive labels. I have it hooked up to the laptop that I keep in the larder, so that I can produce the labels directly once I have packaged up the meat cuts. The feedback from customers, friends and family has been excellent - very professional looking labels that are easy to read.
I sell by piece, rather than by weight, but I still print the weight on the labels where necessary.
I get genuine Brother labels from Amazon at just under a tenner for 400:
https://www.amazon.co.uk/gp/product/B0006HIONW/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1
This is how they look:
Clearly my Hunter ID is normally filled in, as is the Carcase ID from the register I keep on an Excel spreadsheet on the laptop.
In the white box at the bottom I print my name, address and other contact details, whilst for burgers and sausages I also list the full ingredients on the label including the fact that they may contain gluten. This is printed under where this label says "Venison (Muntjac) Haunch". I have since made a few changes, such as including the cull date, but it hopefully gives the general idea.
These are the labels that I shared with the EHO, who was very happy with them.